Large index asymptotics of collision integral kernels of the linear Boltzmann equation in the case of hard-sphere potential

We have identified the asymptotic behavior of the kernels with large indices of integral operators representing the coefficients of expansion in Legendre polynomials of the collision integral of the linear Boltzmann equation for hard-sphere potential. In the case of interacting particles with nonequal absolute values of velocities, the kernels exhibit exponential decay, where the base of the exponent contains the ratio of the lower to higher velocity. In the case of interacting particles with equal absolute values of velocities, the kernels decay according to the power law.